Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-3x-5y &= -2 \\ x-9y &= -6\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = -x-6$ Divide both sides by $-9$ to isolate $y$ $y = {\dfrac{1}{9}x + \dfrac{2}{3}}$ Substitute this expression for $y$ in the first equation. $-3x-5({\dfrac{1}{9}x + \dfrac{2}{3}}) = -2$ $-3x - \dfrac{5}{9}x - \dfrac{10}{3} = -2$ Simplify by combining terms, then solve for $x$ $-\dfrac{32}{9}x - \dfrac{10}{3} = -2$ $-\dfrac{32}{9}x = \dfrac{4}{3}$ $x = -\dfrac{3}{8}$ Substitute $-\dfrac{3}{8}$ for $x$ back into the top equation. $-3( -\dfrac{3}{8})-5y = -2$ $\dfrac{9}{8}-5y = -2$ $-5y = -\dfrac{25}{8}$ $y = \dfrac{5}{8}$ The solution is $\enspace x = -\dfrac{3}{8}, \enspace y = \dfrac{5}{8}$.